Question: Galileo wanted to release a wooden ball and an iron ball from a height of $100$ meters and measure the duration of their fall. He found a plane with an incline of $12^\circ$ that he could climb until he could get to an altitude of $100\text{ m}$. How far should Galileo walk up the inclined plane? Round your final answer to the nearest hundredth.
Answer: The strategy Model the situation as a right triangle. Determine the appropriate trigonometric ratio in order to find the missing side. Form an equation and solve for the missing side. Calculate the final result and round. Modeling as a right triangle This situation can be modeled by the following right triangle. The height is $100\text{ m}$ and the angle on the left is $12^\circ$. We are asked to find how far Galileo should walk up the plane, so we need to find the hypotenuse of the triangle. ${12^{\circ}}$ $?$ $100$ Determining the appropriate trigonometric ratio We are given the measure of an angle and the length of the side ${\text{opposite}}$ to the given angle. We are asked to find the $C{\text{hypotenuse}}$. The appropriate trigonometric ratio is therefore the $\text{sine}$. Forming an equation and solving Denoting the missing side by $x$, we obtain the equation $\sin(12^\circ)=\dfrac{100}{x}$. Solving the equation, we get $x=\dfrac{100}{\sin(12^\circ)}$. Evaluating this result in the calculator and rounding to the nearest hundredth, we get $x=480.97\text{ m}$. Summary Galileo should walk $480.97$ meters up the plane.